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Mirrors > Home > HSE Home > Th. List > hhssims | Structured version Visualization version GIF version |
Description: Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhsssh2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
hhssims.2 | ⊢ 𝐻 ∈ Sℋ |
hhssims.3 | ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
Ref | Expression |
---|---|
hhssims | ⊢ 𝐷 = (IndMet‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssims.3 | . 2 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) | |
2 | hhsssh2.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | hhssims.2 | . . . . 5 ⊢ 𝐻 ∈ Sℋ | |
4 | 2, 3 | hhssnv 29035 | . . . 4 ⊢ 𝑊 ∈ NrmCVec |
5 | 2, 3 | hhssvs 29043 | . . . . 5 ⊢ ( −ℎ ↾ (𝐻 × 𝐻)) = ( −𝑣 ‘𝑊) |
6 | 2 | hhssnm 29030 | . . . . 5 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
7 | eqid 2821 | . . . . 5 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
8 | 5, 6, 7 | imsval 28456 | . . . 4 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) = ((normℎ ↾ 𝐻) ∘ ( −ℎ ↾ (𝐻 × 𝐻)))) |
9 | 4, 8 | ax-mp 5 | . . 3 ⊢ (IndMet‘𝑊) = ((normℎ ↾ 𝐻) ∘ ( −ℎ ↾ (𝐻 × 𝐻))) |
10 | resco 6097 | . . . 4 ⊢ ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) = (normℎ ∘ ( −ℎ ↾ (𝐻 × 𝐻))) | |
11 | 2, 3 | hhssvsf 29044 | . . . . . 6 ⊢ ( −ℎ ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻 |
12 | frn 6514 | . . . . . 6 ⊢ (( −ℎ ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻 → ran ( −ℎ ↾ (𝐻 × 𝐻)) ⊆ 𝐻) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ ran ( −ℎ ↾ (𝐻 × 𝐻)) ⊆ 𝐻 |
14 | cores 6096 | . . . . 5 ⊢ (ran ( −ℎ ↾ (𝐻 × 𝐻)) ⊆ 𝐻 → ((normℎ ↾ 𝐻) ∘ ( −ℎ ↾ (𝐻 × 𝐻))) = (normℎ ∘ ( −ℎ ↾ (𝐻 × 𝐻)))) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ ((normℎ ↾ 𝐻) ∘ ( −ℎ ↾ (𝐻 × 𝐻))) = (normℎ ∘ ( −ℎ ↾ (𝐻 × 𝐻))) |
16 | 10, 15 | eqtr4i 2847 | . . 3 ⊢ ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) = ((normℎ ↾ 𝐻) ∘ ( −ℎ ↾ (𝐻 × 𝐻))) |
17 | 9, 16 | eqtr4i 2847 | . 2 ⊢ (IndMet‘𝑊) = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
18 | 1, 17 | eqtr4i 2847 | 1 ⊢ 𝐷 = (IndMet‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 〈cop 4566 × cxp 5547 ran crn 5550 ↾ cres 5551 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 ℂcc 10529 NrmCVeccnv 28355 IndMetcims 28362 +ℎ cva 28691 ·ℎ csm 28692 normℎcno 28694 −ℎ cmv 28696 Sℋ csh 28699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-lm 21831 df-haus 21917 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-ssp 28493 df-hnorm 28739 df-hba 28740 df-hvsub 28742 df-hlim 28743 df-sh 28978 df-ch 28992 df-ch0 29024 |
This theorem is referenced by: hhssims2 29046 |
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